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Properties

Label 2-546-91.19-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.0459 - 0.998i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + i·3-s + (0.866 + 0.499i)4-s + (0.128 − 0.0344i)5-s + (−0.258 + 0.965i)6-s + (−2.64 + 0.119i)7-s + (0.707 + 0.707i)8-s − 9-s + 0.133·10-s + (3.06 + 3.06i)11-s + (−0.499 + 0.866i)12-s + (1.49 + 3.28i)13-s + (−2.58 − 0.569i)14-s + (0.0344 + 0.128i)15-s + (0.500 + 0.866i)16-s + (−1.67 + 2.89i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + 0.577i·3-s + (0.433 + 0.249i)4-s + (0.0575 − 0.0154i)5-s + (−0.105 + 0.394i)6-s + (−0.998 + 0.0450i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s + 0.0421·10-s + (0.925 + 0.925i)11-s + (−0.144 + 0.249i)12-s + (0.415 + 0.909i)13-s + (−0.690 − 0.152i)14-s + (0.00890 + 0.0332i)15-s + (0.125 + 0.216i)16-s + (−0.405 + 0.701i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0459 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0459 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0459 - 0.998i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0459 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36232 + 1.42635i\)
\(L(\frac12)\) \(\approx\) \(1.36232 + 1.42635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 - iT \)
7 \( 1 + (2.64 - 0.119i)T \)
13 \( 1 + (-1.49 - 3.28i)T \)
good5 \( 1 + (-0.128 + 0.0344i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.06 - 3.06i)T + 11iT^{2} \)
17 \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.85 - 2.85i)T + 19iT^{2} \)
23 \( 1 + (-1.26 + 0.731i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.15 + 3.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.690 + 2.57i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.86 + 6.95i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (11.0 - 2.96i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.78 + 5.65i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.04 + 11.3i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.41 + 5.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.728 - 2.71i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 5.98iT - 61T^{2} \)
67 \( 1 + (-6.01 + 6.01i)T - 67iT^{2} \)
71 \( 1 + (-12.0 - 3.24i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-7.88 - 2.11i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.41 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.93 - 8.93i)T + 83iT^{2} \)
89 \( 1 + (10.6 + 2.84i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.45 + 5.41i)T + (-84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.18107689958042488847354606766, −9.973502121989039093815006655549, −9.484076056997411627820084893291, −8.448202518483514545935767448614, −7.09780978389284852744695956935, −6.41061356145339448703605528229, −5.47934067403427630157764662910, −4.12958950617410481227422678453, −3.68032409217985725627405577060, −2.06498259075962549448759383029, 0.971639237643797032710222894328, 2.83156119281873968215752975627, 3.52287452457173091079341924262, 4.99871352733564928921057334928, 6.15272612488576612052273525678, 6.60471016062180555011349047183, 7.72583705451127984503471909006, 8.870820040957604094547430715927, 9.712666355649536911552307586337, 10.82457568240109263385869227928

Graph of the $Z$-function along the critical line